Finite Group Databasebeta

← Prev (#187) Next (#189) →

C₂×C₃×C₇ #188

order	42
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂²×C₃ (order 12) · Inn(G) = C₁ · Out(G) = C₂²×C₃
holomorph	G ⋊ Aut(G) = C₂×D₃×AGL(1,𝔽₇)
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₄₂
abelian	C₂×C₃×C₇
multiplicative	(ℤ/43)ˣ, (ℤ/49)ˣ, (ℤ/86)ˣ, (ℤ/98)ˣ

Subgroups

	subgroup	G/H
center	G	C₁
commutator, Frattini	C₁	G
LCS	G ▷ C₁
UCS	C₁ ◁ G

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 12
transitive degree	μ_t(G) = 42

A minimal faithful permutation representation — 1 generator on 12 points:

(1,2)(3,4,5)(6,7,8,9,10,11,12)

Conjugacy classes & character table

42 conjugacy classes — view the full character table.

Relationships

center of	C₃×C₇×D₄, C₃×C₇×Q₈, C₃×C₇×Dic₃, …
derived subgroup of	#4245, #4246, #4247, …
Frattini subgroup of	Dic₄₉⋊C₉, C₄×C₄₉⋊C₉, Dic₉×C₄₉, …
Fitting subgroup of	C₇×Dic₃, C₃×Dic₇, Dic₂₁, …
socle of	C₇×Dic₃, C₃×Dic₇, Dic₂₁, …
maximal subgroup of	C₇×Dic₃, C₃×Dic₇, Dic₂₁, …
maximal normal subgroup of	C₇×Dic₃, C₃×Dic₇, Dic₂₁, …
maximal quotient of	C₃×C₄×C₇, C₂²×C₃×C₇, C₂×C₇×C₉, …
automorphism group of	C₄₃, C₄₉, C₂×C₄₃, …
outer automorphism group of	C₄₃, C₄₉, C₂×C₄₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(42, 6);